Contents

Idea

Given a smooth manifoldMM and a vector field X∈Γ(TM)X \in \Gamma(T M) on it, one defines a series of operators ℒX\mathcal{L}_X on spaces of differential forms, of functions, of vector fields and multivector fields. For functions ℒX(f)=X(f)\mathcal{L}_X(f) = X(f) (derivative of ff along an integral curve of XX); as multivector fields and forms can not be compared in different points, one pullbacks or pushforwards them to be able to take a derivative.

For vector fields ℒXY=[X,Y]\mathcal{L}_X Y = [X,Y]. If ω∈Ω•(M)\omega \in \Omega^\bullet(M) is a differential form on MM, the Lie derivativeℒXω\mathcal{L}_X \omega of ω\omega along XX is the linearization of the pullback of ω\omega along the flow exp(X−):ℝ×M→M\exp(X -) : \mathbb{R} \times M\to M induced by XX